According to several references, you might want to use single quotes instead of double ones if they are used inside a quoted portion of text.
Use single quotation marks for quotations within quotations.
Example: He said, "Dan cried, 'Do not treat me that way.' "
Note that the period goes inside both the single and double quotation marks.
If you are using double quotation marks for the “outside quote”, then use single quotation marks for the “inside quote”; if you’re using single quotes on the outside, use doubles on the inside.
“I see no need,” my mother drawled, “for ‘contemplation’, as you put it. Get up off the couch and start doing your homework.”
Annie said, ‘I’ve gone through this whole essay, and I can’t find what your professor means by “that other issue”’.